1. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 J. Bonča Physics Department, FMF, University of Ljubljana, J. Stefan Institute, Ljubljana, SLOVENIA Conductance through coupled quantum dots

2. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Collaborators: R. Žitko, J. Stefan Inst., Ljubljana, Slovenia A.Ramšak and T. Rejec, FMF, Physics dept., University of Ljubljana and J. Stefan Inst., Ljubljana, Slovenia

3. www-f1.ijs.si/~bonca/work.html Cambridge, 2006  Experimental motivation  Single QD: using three different methods: NRG, CPMC and GS – accurate results in a wide parameter regime  DQD system: Large t d : Kondo regimes for odd DQD occupancy Small t d : Two-stage Kondo regime  Three QD’s: Good agreement between CPMC and GS. Two regimes  t’’> G : three peaks in G( d ) due to 3 molecular levels  t’’< G : a single peak in G( d ) of width ~ U At t”<<D, two-stage Kodo effect is found with an unstable non- Fermi liquid fixed point  N-dot system in parallel: RKKY interaction  S=N/2 Kondo effect Introduction

4. www-f1.ijs.si/~bonca/work.html Cambridge, 2006

5. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Double- and multiple- dot structures Craig et el., Science 304, 565 (2004) Holleitner et el., Science 297, 70 (2002)

6. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Quantum Dot (Anderson single impurity problem) d

7. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Quantum Dot d e d +U eded d=e d +U/2 U =1

8. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Quantum Dot d e d +U eded U =1

9. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Quantum Dot d e d +U eded U =1

10. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Quantum Dot d e d +U eded U =1

11. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Quantum Dot d e d +U eded U =1

12. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Quantum Dot d e d +U eded U =1

13. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Quantum Dot d e d +U eded U =1

14. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Quantum Dot d Meir-Wingreen, PRL 68, 2512 (1992) e d +U eded d=e d +U/2 U =1

15. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Quantum Dot d e d +U eded d=e d +U/2 U =1

16. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Quantum Dot d e d +U eded d=e d +U/2 U =1

17. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Quantum Dot d e d +U eded d=e d +U/2 U =1

18. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Quantum Dot d e d +U eded d=e d +U/2 U =1

19. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Quantum Dot d e d +U eded d=e d +U/2 U =1

20. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Quantum Dot d e d +U eded d=e d +U/2 ~ gate voltage U =1 D =U>> G

21. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Three alternative methods:  Constrained Path Monte Carlo method (CPMC), Zhang, Carlson and Gubernatis, PRL 74,3652 (1995);PRB 59, 12788 (1999).  Projection – variational metod (GS), Schonhammer, Z. Phys. B 21, 389 (1975); PRB 13, 4336 (1976), Gunnarson and Shonhammer, PRB 31, 4185 (1985), Rejec and Ramšak, PRB 68, 035342 (2003).  Numerical Renormalization Group using Reduced Density Matrix (NRG), Krishna-murthy, Wilkins and Wilson, PRB 21, 1003 (1980); Costi, Hewson and Zlatić, J. Phys.: Condens. Matter 6, 2519, (1994); Hofstetter, PRL 85, 1508 (2000).

22. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 How to obtain G from GS properties:  CPMC and GS are zero- temperature methods  Ground state energy  Conditions: System is a Fermi liquid N-(noninteracting) sites, N  ∞ G 0 =2e 2 /h Rejec, Ramšak, PRB 68, 035342 (2003) ~ ~

23. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Comparison: CPMC,GS,NRG CPMC, GS-variational, Hartree-Fock: NRG: Meir-Wingreen, PRL 68, 2512 (1992) U<t; Wide-band

24. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Comparison: CPMC,GS,NRG CPMC, GS-variational, Hartree-Fock: NRG: Meir-Wingreen, PRL 68, 2512 (1992) U>>t; Narrow-band

25. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Fermi-liquid E( f ) is a universal function of f Fermi-liquid E( f ) is a universal function of f Number of electrons odd Rejec, Ramšak, PRB 68, 035342 (2003)

26. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Zero-bias conductance Rejec, Ramšak, PRB 68, 035342 (2003)

27. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Rejec, Ramšak, PRB 68, 035342 (2003) Connection of G with charge stiffness

28. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 GS variational method Auxiliary Hamitonian Projection operators Variational wavefunctions: E H : the lowest eigenvalue gives the approximation to the GS of H

29. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 GS variational method – cont. Using Wick’s theorem

30. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Side-coupled Double Quantum Dot PRB 73, 035332 (2006)

31. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Large t d

32. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Large t d – Widths of conductance plateaus: Energies on isolated DQD: d1d1 d2d2

33. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Large t d – Kondo temperatures: Estimating T K using Scrieffer-Wolf:

34. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Large t d – Kondo temperatures: Estimating T K using Scrieffer-Wolf:

35. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Large t d – Adding FM coupling E S=1 E S=0 -J ad

36. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Small t d – Two-stage Kondo effect J eff <T K :Two Kondo temperatures: T K and T K 0 Vojta et al., PRB 65, 140405 (2002); Hofstetter, Schoeller, PRL 88, 016803 (2002), Cornaglia and Grempel, PRB 71, 075305 (2005), Wiel et al., PRL 88, 126803 (2002). Two energy scales: J eff =4t d 2 /U, T K

37. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Small t d – Two-stage Kondo effect J eff <T K :Two Kondo temperatures: T K and T K 0 TKTK TK0TK0 J eff <T K Vojta et al., PRB 65, 140405 (2002); Hofstetter, Schoeller, PRL 88, 016803 (2002), Cornaglia and Grempel, PRB 71, 075305 (2005), Wiel et al., PRL 88, 126803 (2002). Two energy scales: J eff =4t d 2 /U, T K

38. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Small t d – Two-stage Kondo effect J eff >T K J eff 0.250.50 w

39. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Small t d – Two-stage Kondo effect J eff ~T K TKTK 0.250.50 w TK0TK0

40. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Small t d – Two-stage Kondo effect J eff <T K TKTK 0.250.50 w TK0TK0

41. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Small t d – Two-stage Kondo effect J eff <T K ~T TKTK 0.250.50 w Experimental evidence Wiel et al., PRL 88, 126803 (2002).

42. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Large t d – Adding FM coupling Two-stage Kondo effect? Voja et al., PRB 65, 140405 (2002), Hofstetter, Schoeller, PRL 88, 016803 (2002),

43. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Three coupled quantum dots PRB 73, 153307 (2006)  Using CPMC: N CPMC [100,180]  Using GS – variational: N GS [1000,2000]

44. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Three coupled QDs 12 3 Oguri, Nisikawa,Hewson, cond-mat/0504771

45. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 Three coupled QDs Non-Fermi-Liquid NRG Calculation Zitko & Bonca Condmat T K (1) T K (2) TDTD T K (1) T K (2) MO AFM TSK NFL MO AFM

46. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 N - quantum dots PRB 74, 045312 (2006) Schrieffer-Wolff Perturbation in V k 4 -th order

47. www-f1.ijs.si/~bonca/work.html Cambridge, 2006 N - quantum dots  Three different time-scales:  Separation of time-scales:  Different temperature-regimes:

48. www-f1.ijs.si/~bonca/work.html Cambridge, 2006  Using three different methods: NRG, CPMC and GS – accurate results in a wide parameter regime  DQD system: Large t d : Kondo regimes for odd DQD occupancy Small t d : Two-stage Kondo regime  Three QD’s: Good agreement between CPMC and GS. Different phases exist:  t’’> G : three peaks in G( d ) due to 3 molecular levels  t’’< G : a single peak in G( d ) of width ~ U  Two-stage Kondo regime, when t’’<T K  NFL behavior is found  N-dot system in parallel: RKKY interaction Conclusions